Leokadia Bialas-Ciez

Jackson's inequality in the complex plane and the Łojasiewicz-Siciak inequality of Green's function

We prove a generalization of Jacksons inequality for compact sets in the complex plane admitting both upper and lower bounds for their Greens functions, i.e.?the well known Holder Continuity Property (HCP) and the less known but crucial Łojasiewicz-Siciak inequality (ŁS). Moreover, we show that ŁS is a necessary condition for our Jackson type inequality. The resulting Jackson Property provides us with a large class of sets in the complex plane for which the global and local Markov inequalities are equivalent.

On Lagrange polynomials and the rate of approximation of planar sets by polynomial Julia sets

Abstract We revisit the approximation of nonempty compact planar sets by filled-in Julia sets of polynomials developed in [27] and analyze the rate of approximation. We use slightly modified fundamental Lagrange interpolation polynomials and show that taking certain classes of nodes with subexponential growth of Lebesgue constants improves the approximation rate. To this end we investigate properties of some arrays of points in C . In particular we prove subexponential growth of Lebesgue constants for pseudo Leja sequences with bounded Edrei growth on finite unions of quasiconformal arcs. Finally, for some classes of sets we estimate more precisely the rate of approximation by filled-in Julia sets in Hausdorff and Klimek metrics.